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If a planet has a period of 8 years, what is its average distance from the Sun, in astronomical...

Question:

If a planet has a period of 8 years, what is its average distance from the Sun, in astronomical units?

Assume that the mass of the planet is small compared to the Sun and that the average distance from the Sun is the same as the semi-major axis.

(Express your answer with at least one digit after the decimal point.)

Planetary Motion

Kepler deduced three empirical laws regarding planetary motion by analyzing the meticulous and voluminous observational data regarding planetary positions, gathered by Tycho Brahe over a time span of twenty years. In turn, these laws formed the basis for Newton's theory of gravitation. According to Kepler's third law, the time period of orbit of a planet around the sun and the length of the semi-major axis of the elliptic orbit are related according to a law similar in form to that relating the time period of oscillations of a musical instrument to the length of the oscillating structure in it. Hence it was called the Law of Harmonies.

Answer and Explanation:

According to Kepler's third law, the time period of orbit of a planet around the sun and the length of the semi-major axis are related according to,

{eq}\displaystyle {T^2\propto a^3} {/eq}

Hereafter it is assumed that the length of the semi-major axis is equal to the average distance from the sun. From the above proportionality if follows that for two different planets with time periods {eq}\displaystyle {T_1} {/eq} and {eq}\displaystyle {T_2} {/eq} and average distances {eq}\displaystyle {a_1} {/eq} and {eq}\displaystyle {a_2} {/eq},

{eq}\displaystyle { \dfrac{T_2^2}{T_1^2}=\dfrac{a_2^3}{a_1^3}}---------(1) {/eq}

Now we know that for the earth the time period is one year and the average distance is 1 astronomical unit. For the unknown planet the time period is given to be 8 years. Using these data in (1) we get,

{eq}\displaystyle { \dfrac{8^2}{1^2}=\dfrac{a_2^3}{1^3}} {/eq}

Therefore,

{eq}\displaystyle {\boxed{a_2=4.0\ AU}} {/eq}.


Learn more about this topic:

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Kepler's Three Laws of Planetary Motion

from Basics of Astronomy

Chapter 22 / Lesson 12
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